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how to teach your child numbers arithmetic mathematics

understanding graphs and charts

There are all manner of ways in which you can illustrate and present data in order to make it easier for yourself and others to understand. The terms ‘charts’ and ‘graphs’ are often used interchangeably or sloppily, but some people like to reserve the term ‘graphs’ for diagrams with numbered axes and for continuous scales.

The choices of graphs and charts, and the scales used on those illustrations, mean it is very easy to mislead and confuse. So graphs and charts are widely popular with all manner of con artists and swindlers.

You see this number line (from minus and zero, dealing with no-thing and less than nothing!) is counting forward and back from zero.

And from fractions, decimals and percentages 2, you can see numbers counting up from zero in two different directions. As you see in this second diagram, the horizontal line, or axis, is now marked with an X and the vertical axis is now marked with a Y.

straight line and parabola graphs

Now a very clever fellow named René Descartes [1596-1650] worked out that he could represent all manner of sums by expanding on these ideas. We extend both these axes into the negative field.

We can plot values for different equations and draw the lines, whether straight or curved, involved.

Here is a graph plotting y = x. The plotted points that are joined to make the line of the graph are in white.
	The points plotted are:
	x	y
	-4	-4
	-3	-3
	-2	-2
	-1	-1
	0	0
	1	1
	2	2
	3	3
	4	4

Here is a graph plotting y = 2x. The plotted points that are joined to make the line of the graph are in white.
	The points plotted are:
	x	2x (=y)
	-4	-8
	-3	-6
	-2	-4
	-1	-2
	0	0
	1	2
	2	4
	3	6
	4	8

Here is a graph plotting y = x².	The points plotted are:
	x	x² (=y)		x	x² (=y)
	-10	100		1	1
	-9	81		2	4
	-8	64		3	9
	-7	49		4	16
	-6	36		5	25
	-5	25		6	36
	-4	16		7	49
	-3	9		8	64
	-2	4		9	81
	-1	1		10	100
	0	0
Note that this graph line is equivalent to x = √y.

Here is a graph plotting y =2n, y =3n and y = 4n	The points plotted are:
	n	2n	3n	4n
	-4	1/16
	-3	1/8	1/27
	-2	1/4	1/9	1/16
	-1	1/2	1/3	1/4
	0	1	1	1
	1	2	3	4
	2	4	9	16
	3	8	27
	4	16
Note: y =2-n is the same as saying 1/2n.

In this last set of graphs, note that each line crosses the y axis at y = 1 when n = 0.

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pie charts

Pie charts are the most basic type of graph, used to illustrate disparate collections of objects, such as how many apples, oranges and children are there. These are collections separated by quality.

The complete pie chart holds 100% of the items.
The size of each section is worked out as a percentage, that is as how many hundredths of the whole.

Thus, if there are 12 children, 5 apples and 7 oranges, the whole pie is made up of 12 + 5 + 7, or 24, objects.
The children make half the pie, or 50% (50/100 or 12/24), the apples are 5/24 of the pie or 28% [(5 ÷ 24) x 100], while the oranges are 7/24 of the pie or 32% [(5 ÷ 24) x 100].
Note that you can check this sum by adding the percentages to make sure they do make 100%: 50% + 28% + 32% = 100%.

bar charts

A bar chart has broad lines, or bars, whose lengths are proportional to the values that they represent. The greater the height or length, the greater the value. The bars can go horizontally or vertically, and can be drawn to appear three-dimensional.

In comparison with the planets, the radius of the Sun is 685,000 km, almost ten times that of Jupiter.

When only a small portion of one or both of the scales are used, often the axis/axes are ‘abbreviated’ by omitting part of the scale, either near the zero mark or further along the axis. This shown by inserting a zigzag,

or a break to mark the portion of the scale that is omitted.

other types of graph available

Using different graph paper, it is possible to plot special graphs, such as ones with a logarithmic scale. here is such a graph (click on the image to go to a full-size version at Is Intelligence Distributed Normally? By Cyril Burt, 1963).

Plotting I.Q. against proportion in the population

graphs indicating uncertain data

To show how uncertain data is graphed, we are using graphs that illustrate global temperatures. Measuring the whole world’s temperature is necessarily problematic, with temperatures being taken by many people over centuries, in many places - sea, land and air - with differing instruments and methods. Standardising such data can only be achieved to a level of confidence, a probability of accuracy, as is illustrated just below.

In the next graph, you will notice that each year’s temperature bar estimate [in red] is overlaid by what looks like a long letter ‘I’ [in black]. That usually indicates that the probability is that we have a 95% ‘confidence’ that the real temperature fell between the two cross bars of the ‘I’. Notice that as you go further back in time, the range of uncertainty (the vertical bar of the ‘I’) increases for obvious reasons.

The blue line is a typical moving average, smoothing out natural variations.

Ocean temperature data using NOAA (ocean) satellites system,
calibrated by CRU ground-based temperature measurements

Sometimes, you will see such ranges of uncertainty indicated by ‘fans’, often giving more than one uncertainty level. This is especially common in graphs giving future forecasts in complex domains, like economics and climatology.

Solid lines are multi-model global averages of surface warming (relative to 1980–1999) for the scenarios A2, A1B and B1, shown as continuations of the 20th century simulations.
Shading denotes the ±1 standard deviation range of individual model annual averages.
The orange line is for the experiment where concentrations were held constant at year 2000 values.
The grey bars at right indicate the best estimate (solid line within each bar) and the likely range assessed for the six SRES marker scenarios. The assessment of the best estimate and likely ranges in the grey bars includes the AOGCMs in the left part of the figure, as well as results from a hierarchy of independent models and observational constraints.

Notice carefully how the line narrows to a single point at about the year 2000. This is our present day best estimate. Notice also how the fans spread as they go both forwards and backwards [in grey] in time, with our past estimate and with our future forecasts becoming more and more uncertain the more distant the time.

Here is another type of fan graph. Graphs on economic forecasting are particularly interesting because the numbers are ever changing their real meaning as governments manipulate inflation. The forecasts are commonly heavily politically loaded. For example, governments facing an election, and companies trying to sell shares, regularly project growth far above any realistic level, and other parties make self-interested ‘forecasts’. Thus, it is hardly surprising that a great number of economic forecasts turn out to be false.

source: bankofengland.co.uk The fan chart depicts the probability of various outcomes for CPI [Consumer Price Inflation] inflation in the future.
If economic circumstances identical to today’s were to prevail on 100 occasions, the MPC’s [Monetary Policy Committee] best collective judgement is that inflation over the subsequent three years would lie within the darkest central band on only 10 of those occasions.
The fan chart is constructed so that outturns of inflation are also expected to lie within each pair of the lighter red areas on 10 occasions. Consequently, inflation is expected to lie somewhere within the entire fan charts on 90 out of 100 occasions. The bands widen as the time horizon is extended, indicating the increasing uncertainty about outcomes.

reading graphs - don’t be hoodwinked!

Note that many graphs either do not mark that a graph scale does not start from zero, or do not do so obviously. This makes it easy to misinterpret data being provided on such a graph.

“Most official United Kingdom government publications ignore the need to mark axes which do not start from zero, as do most graphs in newspapers, thereby greatly diminishing their value. Without such an indication it will appear to readers that the distance of the graph from the horizontal axis is proportional to the data being displayed; the impression given by the graph will greatly exaggerate changes over time unless it is made abundantly clear that the vertical scale does not start from zero.” [Quoted from Royal Statistical Society .pdf]

Here are two graphs that illustrate how graphs can be used to mislead. The graph on the left is drawn confusingly, even though displayed on a supposedly reputable web-site. The graph on the right shows another way of showing closely related data to the left graph, but with an extra point added before and after, thus presenting a very different picture. An explanation of the changes.

data source: Global Warming Policy Foundation

data source: Global Warming Policy Foundation graph source: timesonline.typepad.com (Note that the legend,“Mean Global Temperature (C)” should read “MGT Index”, 14° must be added to the Index to obtain the MGT [Mean Global Temperature]”)

Notice that the graph to the right, the left-hand, y-axis is stretched relative to the equivalent axis on the left-hand graph. This has the visual effect of making the changes look much bigger on the right-hand graph, or much smaller on the left-hand graph.

Another trick:
On the left-hand graph, the actual temperatures in degrees Centigrade are given. On the right-hand graph, the variations in temperature are given; but it is essentially the same data on both graphs. Notice that on the left-hand graph, the temperatures between 0 and 14.3°C are not marked on the axis and, of course, neither are the temperature es from -273°C (0°K, absolute zero temperature). Strictly, this missing part of the axis would be marked with a zigzag or a break.

What matters is whether a graph is clear and does not mislead (of course, it is easy to be misled if you do not have much experience of reading graphs). Temperatures below 0°C do not tend to interest humans most of the time, unless they are Eskimos/Inuit. It is not misleading to concentrate on temperature variations when considering global warming, though maybe it is misleading when discussing the temperature of the sun, or when making ice cubes - context is always relevant.

Note that the left graph is illustrating global temperatures, a phenomenon which has short-term variations and a long-term trend. By showing data for just eight years, instead of centuries, a very misleading notion is being given of global temperature change. In fact, the long-term trend is that global temperatures are increasing, as can be seen in this graph above.

another misleading graph

Here, the date ranges become smaller as the periods approach the furthest future, giving impression of decreasing emissions. abelard.org has added the period lengths in years to make clear the obfuscation by Climate Brief. Note, on the original page, the graph redraws continually, making it conveniently difficult to follow what is being presented.

denying anthropogenic global warming - a couple of amusing graphs

First, here is a graph as annotated by global warming sceptics. Supposed trend lines (in blue) have been added to imply that temperatures are falling.
The sceptics carefully chose a higher point and a lower point from among the varying data, in order to produce a line that varies downwards.

Skeptics' climate 'graph'

Next, a mathematically sane trend line (in red) over the full chosen data set, that shows a steady increase in temperature.
Of course, this data set also starts at a selected point.

Realistic graph of global warming

[For further discussion on AGW, see anthropogenic global warming, and ocean acidity.]

another example from the fossil media: the groaniad and visual lies

[Note: GMG = Guardian Media Group]

GMG plc’s revenues as depicted in their annual report:

Observe how GMG have put the latest year at the top of the graph, when it is usual to ‘read’ downwards.

And here is the same data re-arranged by Guido Fawkes:

“Guido has re-jigged the chart to make things clearer, the traditional bar-chart shows revenue has dropped by a further third since 2008.”

innumerate 'reporters' and helicopters - how a previous prime minister obfuscated

Gordon Brown the Clown apparently says:

"There had been a huge funding increase and helicopter numbers have increased by 60 per cent in the last 3 years."

There is, of course, a 117% probability he is lying again.

Why am I not hearing the actual and real number of 'extra' helicopters over three years?

So, I am going to guess. There were ten helicopters, three years ago, and now there are sixteen.
Or perhaps there were five, and now there are eight.

Why are we being given percentage claims instead of numbers?

Why are we not being told how many helicopters the operations in Afghanistan requires?

But more to the point, why are our innumerate reporters supinely accepting this blather?

And where are these helicopters coming from? Iraq?

There is usually only one reason to quote percentages without base numbers, and that reason is base dishonesty.

end notes

1. Continuous
   For instance, measuring stick, clock, no limit to subdivision, the measurement of continuous space and time, the arithmetic expressions 1·84, 1·333 recurring, 247, 247·838, 4701·367294, etc.
   
   This is not the counting of supposedly separated objects, but is comparing one object with another, for example by using a length of stick with marks on called a ruler.
   
   It is commonplace to confuse the numbers used to count separated ‘objects’ with the numbers used to indicate a place upon a ruler or clock. Note that the number 247, in Section 3.above, looks very like the number 247 in this section, but they are being used in very different ways. In Section 3, ‘the’ number is being used to count objects; in Section 4, to mark a position.
   [ Extract from Gödel and sound sets, Metalogic A1]
   
   
2. Axes is the plural of the word axis. So axes means “more than one axis”.
   
   
3. Strictly speaking, there is no line joining the dots but, of course, you can put more points in, such as 2·1, 2·25, 2·5 and so on. You can put as many points as you wish. Put enough dots in and they will, in fact, join up into a line. If you make your dots fine enough, it will take more dots before you cannot see the spaces between the dots.

   This becomes difficult for theorists, and even very complicated. Further discussion of these matters come as you advance in age and in wisdom. However, it is important at this point to be aware of the difference between the actual points that you plot on a graph and the sometimes rather imaginary artistic lines you may use to join the dots. Make sure your graph gives an honest and meaningful story.
   
   

4. “Mr Ward, flagged up the graph because the shape looked different to any of the standard global temperature patterns over this period. In particular, it appeared to show a stronger cooling trend. In an email to Dr Peiser he wrote:

   "It does not appear to correspond to any of the official records of global average temperature published by the Met Office, NASA or NOAA and not even that of the University of Arizona at Huntsville. I am particularly intrigued by the graph's indication that 2003 was a warmer year than 2005, which none of the official records show."

   “In response, Dr Peiser admitted there had been a "small error", which resulted in the 2003 temperature being plotted as cooler rather than hotter than 2005.”
   —
   “The GWPF used a figure of 0.487 instead of 0.457 as the temperature for 2003. ” [Quoted from timesonline.typepad.com]

   It’s wonderful what you can do with graphs if you are sufficiently creative.
   

5. Climate brief received several hundred thousand pounds funding from a major NGO, the European Climate Foundation. "ECF is one of the most influential NGO's in the European climate debate. It it also true that it has ample funds – it was able to spend €26 million in 2013..." They wish to " advance discussion on decarbonisation of the power sector and the broader economy in Europe".

email email_abelard [at] abelard.org

© abelard, 2009, 28 october

all rights reserved

the web address for this page is https://www.abelard.org/sums/teaching_number_arithmetic_mathematics_understanding_graphs_charts.php

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